Operations in Milnor K-theory

We show that operations in Milnor K-theory mod p of a field are spanned by divided power operations. After giving an explicit formula for divided power operations and extending them to some new cases, we determine for all fields k(0) and all prime numbers p, all the operations K-i(M)/p -> K-j(M)/p commuting with field extensions over the base field k(0). Moreover, the integral case is discussed and we determine the operations K-i(M)/p -> K-j(M)/p for smooth schemes over a field. (C) 2008 Elsevier B.V. All rights reserved.